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Re: value of g in black holes (long)



Re: value of g in black holes (long)

Regarding Dwight's question:

I'm currently going over gravitational forces and acceleration with
my first
year physics students. Of course, one of the topics that the students'
had
brought up were black holes. One of the questions, I didn't know and
haven't been
able to find the answer...so I'm coming to the experts! :) What must
be the
value of g (in m/s/s) so that a "black hole" would form?

Thanks
Dwight Souder
Ashland, OH

It's not so much a matter of a particular g value (say at the
surface) as it is a matter of the failure of a gravitating object's
internal bulk modulus to continue to accommodate the relentless
increase in the gravitationally induced internal *pressure* (due to
the weight of the overlying layers) in the body with a continued
decrease in radius. The precise point where the object's internal
bulk rigidity is insufficient to support the overlying weight depends
on the details of the equation of state of the material composing the
object and its overall mass.

In the case of the nuclear matter of a neutron star the Oppenheimer-
Volkoff limit of about 3 solar masses is the largest mass a stable
neutron star can have without collapsing under its own weight to a
black hole. The main source of support for such nuclear matter is
neutron degeneracy pressure. Even though this nuclear matter is
extremely stiff with an enormous bulk modulus (recall a collapsing
massive star bounces off of a neutron nuclear matter core in the
midst of a supernova reversing an implosion into an explosion more
luminous than a galaxy without significantly compressing the core
even though the change in velocity of the bouncing matter is
comparable to c) gravitation is *still* capable of crushing nuclear
matter once the mass reaches a critical value.

This is because in GR it is possible for the central pressure in a
dense body to diverge to infinity with a *finite* amount of mass
having a *finite* density. The reason for this is a positive
feedback effect that occurs in GR (absent in Newtonian gravity)
arising from a combination of the intrinsic nonlinearity of the
equations of GR and fact that in GR gravitation has sources other
than just mass (as is the case in Newtonian gravity). In GR
mass/energy, momentum, and stress of all kinds act as sources for
producing gravity (i.e. spacetime curvature). Hydrostatic pressure
is a form of stress; as such *it* produces gravity. So when a
compact body becomes so dense that GR effects become significant for
it the gravitational field around that body is produced by a
combination of the body's mass *and* its internal pressure. At
sufficient density a tiny increase in compression causes an increase
in g, and this increases the weight of the overlying mass, and this
increases the interior pressure. The increased pressure causes even
more gravitation, thus further increasing g and this cause further
increase in weight and pressure, etc. At a certain critical mass and
internal density, the central pressure inexorably increases rapidly
toward infinity. So no matter how stiff the interior matter is, the
inward gravitational compressive forces are guaranteed to eventually
overwhelm the material's stiffness, and the interior of the body will
begin to collapse inward once the material stiffness fails to keep up
with the increasing pressure.

In the case of the nuclear matter of a neutron star, the nucleons
begin too be crushed into each other causing the material to become
a soup of quark matter. Once the quarks get sufficiently close
together asymptotic freedom weakens the Strong force between them.
In short order the quark matter turns into a mass of *free*
noninteracting quarks whose only repulsive tendency is due to
their quantum degeneracy. At this point the inward gravitational
forces are so much higher than the quark degeneracy pressure that the
quarks fail to put up much of a fight at all (relative to the
gravitational forces) and the whole thing collapses as if it were
effectively in free fall.

I have never looked into the matter but I have heard of proposals
of the possibility of 'quark stars' where the central core is
supposed to composed of a stable soup of quark matter. I don't know
about the actual possible viability of such a system, but I expect
that if such a thing is really possible, it could only exist within a
narrow range of masses quite close to the Oppenheimer-Volkhoff limit
for the maximum mass of a neutron star.

If we crudely assume that the Oppenheimer-Volkhoff limit is exactly 3
solar masses and crudely assume that the matter of a neutron star has
a *uniform* (nearly) incompressible bulk nuclear matter of density
2.8 x 10^17 kg/m^3 we can calculate the central pressure of a static
spherically symmetric distribution of such matter near its yield
limit. The result is a pressure of about 2.4 X 10^34 Pa at the point
where nuclear matter begins to fail to support overlying gravitating
matter. With this limiting model the corresponding surface gravity g
becomes 5.6 x 10^12 m/s^2. Also this limiting case has a radius (in
Standard coordinates) of 12 km which is 1.34 times the system's
Schwarzschild radius.

Since any gravitating body has its density monotonically increase
with depth below the surface, and since gravitational forces increase
strongly with decreasing separation, we can use these facts to place
a strict *lower* bound on the central pressure in a spherically
symmetric distribution of gravitating matter. Suppose we have a
spherically symmetric static distribution of matter that has a fixed
given radius and a fixed total mass. Of all possible ways of
spherically distributing this matter within the available volume so
that the density is nondecreasing with depth we can show that the
pressure at the center is *minimum* for a mass distribution which has
*uniform* density. If the density actually increases with depth (as
in a realistic case) then the central pressure must be higher than
that for a uniform mass distribution of an incompressible substance
of the same mass and same overall volume. This is convenient because
we can actually exactly solve using GR for the central pressure P(0)
in the center of an isolated sphere of an incompressible fluid of
some given mass density [rho] and overall radius R (in standard
coordinates). The result is:

P(0) = ([rho]*c^2)*(1 - A)/(3*A - 1)

where A == sqrt(1 - r_s/R) and r_s == 2*G*M/c^2 which is the
Schwarzschild radius. Notice that as A --> 1/3 the value of P(0)
diverges to infinity. This limiting case occurs when R/r_s --> 8/9.

The above considerations tell us that there is a strict lower bound
on the radius of a given spherically symmetric distribution of a
fixed amount of matter that is statically stable against
gravitational collapse. In particular, we know that the radius of
the object *must* be *at least* (9/8)*r_s = (9/4)*G*M/c^2 (simply
because no realizable equation of state for real matter can withstand
an actual infinite pressure without collapse).

In addition, we can get an *upper* bound for the surface gravity g
for the object of g_max = (16/27)*(c^4)/(G*M). Notice that the
maximum possible surface gravity is inversely proportional to the
total mass present.

We can calculate the gravitational redshift z (observed at infinity)
for light emitted from the surface of a spherically symmetric object.
The result is z = 1/A - 1 . Thus when A approaches its limiting
value of 1/3 we see that z approaches 2. This puts a strict upper
bound of z_max = 2 on the observed gravitational redshift for
radiation emitted from the surface of a spherical body. So when we
see distant quasars whose redshifts exceed 2 we know that that
redshift cannot be gravitational in origin, but must be cosmological
in nature (since the only other alternative of it being a
systematic recessional Doppler effect is ruled out by invoking the
Copernican Principle). This is one reason why astronomers consider
quasars to be very distant objects rather than relatively close by
ones with a huge gravitational redshift (or close by ones that all
just happen to be running away from us very fast).

I know Weinberg's book _Gravitation_and_Cosmology_ discusses this
limiting radius. I haven't checked other books like MTW & others
to see if it is in them too.

David Bowman
David_Bowman@georgetowncollege.edu